1. Jordan blocks

Dear MHF members,

my problem reads as follows. I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.
Problem. Consider the system $\displaystyle \mathbf{X}^{\prime}=\mathbf{A}\mathbf{X}$, where $\displaystyle \mathbf{A}$ is a $\displaystyle 30\times30$ constant matrix with only two distinct eigenvalues $\displaystyle \lambda_{1}=1$ and $\displaystyle \lambda_{2}=2$ Suppose that the Jordan form $\displaystyle \mathbf{J}$ of $\displaystyle \mathbf{A}$ has Jordan blocks $\displaystyle \mathbf{J}_{3},\mathbf{J}_{3},\mathbf{J}_{6}$ corresponding to $\displaystyle \lambda_{1}$ and blocks $\displaystyle \mathbf{J}_{1},\mathbf{J}_{2},\mathbf{J}_{3},$$\displaystyle \mathbf{J}_{3},\mathbf{J}_{4},\mathbf{J}_{5}$ corresponding to $\displaystyle \lambda_{2}$, where $\displaystyle \mathbf{J}_{k}$ is a $\displaystyle k\times k$ Jordan block.

1. How many linearly independent eigenvectors does $\displaystyle \mathbf{A}$ have?
2. Suppose that $\displaystyle \mathbf{V}$ is not an eigenvector but satisfies $\displaystyle (\mathbf{A}-\lambda_{2}\mathbf{I})^{3}\mathbf{V}=0$. How many linearly independent solution vectors $\displaystyle \mathbf{V}$ are there?
3. Suppose that the Jordan blocks occur down the diagonal of the $\displaystyle \mathbf{J}$ in the order stated above. Which matrix elements of $\displaystyle \lim_{t\to\infty}t^{4}\mathrm{e}^{-\lambda_{2}t}\mathrm{e}^{\mathbf{J}t}$ are non-zero and what are their values? $\displaystyle \rule{0.2cm}{0.2cm}$

Thanks a lot.
bkarpuz

2. Hint for 1

$\displaystyle \dim \ker (J_k-\lambda I_k)=k-\textrm{rank}(J_k-\lambda I_k)=k-(k-1)=1$

Originally Posted by bkarpuz
I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.

Can you comment those ideas?

3. Originally Posted by FernandoRevilla
Can you comment those ideas?
1. The number of regular eigenvectors are $\displaystyle 9$, and the remaining $\displaystyle 21$ are generalized eigenvectors.
2. I feel like it is $\displaystyle 15$, but I really need help with this one.
3. The terms of the blocks of the exponential matrix corresponding to $\displaystyle \lambda_{1}=1$ will all vanish. For $\displaystyle \lambda_{2}=2$, all the terms except one will vanish. The non-zero term is located at $\displaystyle (26,30)$, and its value is $\displaystyle 1/4!$.

4. Originally Posted by bkarpuz
1. The number of regular eigenvectors are $\displaystyle 9$, and the remaining $\displaystyle 21$ are generalized eigenvectors.

Right.

2. I feel like it is $\displaystyle 15$, but I really need help with this one.

Right. Denote $\displaystyle J_{k,\lambda_i}$ a Jordan block of order k associated to the eigenvalue $\displaystyle \lambda_i$. Then,

$\displaystyle (i)\quad \textrm{rank} \begin{bmatrix}{J_{3,\lambda_1}-\lambda_2 I_3}&{0}&{0}\\{0}&{{J_{3,\lambda_1}-\lambda_2 I_3}&{0}\\{0}&{0}&{{J_{6,\lambda_1}-\lambda_2 I_6}\end{bmatrix}^3=3+3+6=12$

$\displaystyle (ii)\quad \textrm{rank}(J_{k,\lambda_2}-\lambda_2I_k)^3=\begin{Bmatrix}{ 0}&\mbox{ if }& k=1,2,3\\1 & \mbox{if}& k=4\\2 & \mbox{if}& k=5\end{matrix}$

According to the canonical form structure, we verify

$\displaystyle \dim (\ker (A-\lambda_2I_{30})^3)=30-\textrm{rank}(A-\lambda_2I_{30})^3=30-15=15$

3. The terms of the blocks of the exponential matrix corresponding to $\displaystyle \lambda_{1}=1$ will all vanish. For $\displaystyle \lambda_{2}=2$, all the terms except one will vanish. The non-zero term is located at $\displaystyle (26,30)$, and its value is $\displaystyle 1/4!$.

Right. The non zero term corresponds to the block

$\displaystyle J_{5,\lambda_2}$